Abstract

Suppose that is a real normed linear space, is a nonempty convex subset of , is a Lipschitzian mapping, and is a fixed point of . For given , suppose that the sequence is the Mann iterative sequence defined by , where is a sequence in [0, 1], , . We prove that the sequence strongly converges to if and only if there exists a strictly increasing function with such that .

1. Introduction

Let be an arbitrary real normed linear space with dual space , and let be a nonempty subset of . We denote by the normalized duality mapping from to defined by
where denotes the generalized duality pairing.

A mapping is called strongly pseudocontractive if there exists a constant such that, for all , there exists satisfying
is called -strongly pseudocontractive if there exists a strictly increasing function with such that, for all , there exists satisfying
is called generalized -pseudocontractive (see, e.g., [1]) if there exists a strictly increasing function with such that
holds for all and for some .

Let denote the fixed point set of . If , and (1.3) and (1.4) hold for all and , then the corresponding mapping is called -hemicontractive and generalized -hemicontractive, respectively. It is well known that these kinds of mappings play important roles in nonlinear analysis.

-hemicontractive (resp., generalized -hemicontractive) mapping is also called uniformly pseudocontractive (resp., uniformly hemicontractive) mapping in [2, 3]. It is easy to see that if is generalized -hemicontractive mapping, then is singleton.

It is known (see, e.g., [4, 5]) that the class of strongly pseudocontractive mappings is a proper subset of the class of -strongly pseudocontractive mappings. By taking , where is a strictly increasing function with , we know that the class of -strongly pseudocontractive mappings is a subset of the class of generalized -pseudocontractive mappings. Similarly, the class of -hemicontractive mappings is a subset of the class of generalized -hemicontractive mappings. The example in [6] demonstrates that the class of Lipschitzian -hemicontractive mappings is a proper subset of the class of Lipschitzian generalized -hemicontractive mappings.

It is well known (see, e.g., [7]) that if is a nonempty closed convex subset of a real Banach space and is a continuous strongly pseudocontractive mapping, then has a unique fixed point . In 2009, it has been proved in [8] that if is a nonempty closed convex subset of a real Banach space and is a continuous generalized -pseudocontractive mappings, then has a unique fixed point .

Theorem XCZ (see [6, Theorem 3.2]). Let be a real normed linear space, let be a nonempty convex subset of , and let be a Lipschitzian generalized -hemicontractive mapping. For given , suppose that the sequence is the Mann iterative sequence defined bywhere is a sequence in satisfying the following conditions: (1),
(2). Then converges strongly to the unique fixed point of in .

The main purpose of this paper is to give necessary and sufficient condition for the Mann iterative sequence which converges to a fixed point of general Lipschitzian mappings in an arbitrary real normed linear space. As an immediate consequence, we will obtain necessary and sufficient condition for the Mann iterative sequence which converges to a solution of a general Lipschitzian operator equation .

2. Preliminaries

The following lemmas will be used in the proof of our main results.

Lemma 2.1 (see, e.g., [12]). Let be a real normed linear space. Then for all , we have

Lemma 2.2 (see, e.g., [13]). Let , , be three nonnegative sequences satisfying the following condition:
where is some nonnegative integer, , and . Then the limit exists.

Lemma 2.3. Suppose that is a strictly increasing function with and there exists a natural number such that , , , and are nonnegative real numbers for all satisfying the following conditions: (i),
(ii),
(iii). Then .

Proof. Without loss of generality, let . Now, we will show that . Consider its contrary: or . For any given , there exists a nonnegative integer such that and for all . By condition (i), we have
Using Lemma 2.2 and condition (ii), we obtain that exists and is bounded. Suppose that , where is a nonnegative constant. It follows that
Thus,
which is a contradiction. Therefore,
By condition (ii), for all , there exists a nonnegative integer such that
By (2.6), there exists a natural number such that . Now, we prove the following inequality (2.8) holds for all :
It is obvious that (2.8) holds for . Assuming (2.8) holds for some , we prove that (2.8) holds for . Suppose this is not true, that is, . Then and so . Noting that , it follows from condition (i), (2.7), and (2.8) that
which is a contradiction. This implies that (2.8) holds for . By induction, (2.8) holds for all . From (2.7), and (2.8), we have
Taking , we obtain . By (2.6), we have . This completes the proof.

Remark 2.4. Lemma 2.3 is different from Lemma 3 in [14], which requires that for all . It is also different from Lemma 2.3 in [6], which requires that .

3. Main Results

Theorem 3.1. Let be a real normed linear space, be a nonempty convex subset of , let be a Lipschitzian mapping, and let be a fixed point of . For given , suppose that the sequence is the Mann iterative sequence defined by
where is a sequence in satisfying the following conditions: (i), (ii). Then converges strongly to if and only if there exists a strictly increasing function with such that

Proof. First, we prove the sufficiency of Theorem 3.1.Suppose there exists a strictly increasing function with such that (3.2) holds. Let
Then there exists such that
By (3.2), we obtain . Taking , then
From (3.1) and (3.4), by using Lemma 2.1, we obtain
where is the Lipschitzian constant of . It follows from (3.1) that
Substituting (3.7) into (3.6), we have
Setting , it follows from (3.8) that
It follows from condition (i) that . Thus, there exists a natural number such that for all . Let
Since for all , by (3.9) and (3.10), we have
It follows from condition (i) that . Therefore, by (3.5), condition (ii), and Lemma 2.3, we obtain that . That is, converges strongly to .Finally, we prove the necessity of Theorem 3.1.Assume that converges strongly to . Let be the Lipschitzian constant of . For all , we have
Taking , then is a strictly increasing function with , and . From (3.12), we obtain
which implies (3.2) holds. This completes the proof of Theorem 3.1.

Remark 3.2. If is a generalized -hemicontractive mapping, then (3.2) holds. By Theorem 3.1, we obtain Theorem XCZ.

Theorem 3.3. Let be a real Banach space, let be a nonempty closed convex subset of , and let be a Lipschitzian generalized -pseudocontractive mapping. For given , suppose that the sequence is the Mann iterative sequence defined by
where is a sequence in satisfying the following conditions: (1),
(2). Then converges strongly to the unique fixed point of in .

Proof. By Theorem 2.1 in [8], has a unique fixed point in . By Theorem 3.1, converges strongly to . This completes the proof of Theorem 3.3.

Theorem 3.4. Let be a real normed linear space, let be a Lipschitzian operator, and let and be a solution of the equation . For given , suppose that the sequence is the Mann iterative sequence defined by
where is a sequences in satisfying the following conditions: (i), (ii). Then converges strongly to if and only if there exists a strictly increasing function with such that

Proof. Define by . Since , we have . From (3.15), we obtain . Since
Therefore,
The condition (3.16) is equivalent to condition (3.2). Since is a Lipschitzian operator, is a Lipschitzian mapping. By Theorem 3.1, Theorem 3.4 holds. This completes the proof of Theorem 3.4.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (NSFC) (Grant no. 71201093, no. 11001289, and no. 11171363), Humanities and Social Sciences Foundation of Ministry of Education of China (Grant no. 10YJCZH217), Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (Grant no. BS2012SF012), and Independent Innovation Foundation of Shandong University, IIFSDU (Grant no. 2012TS194).